The Stacks project

Lemma 65.49.3. Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of algebraic spaces over $S$. Then for $x \in |X|$ we have: $x$ has codimension $0$ in $X \Rightarrow f(x)$ has codimension $0$ in $Y$.

Proof. Via Properties of Spaces, Lemma 64.11.1 and ├ętale localization this translates into the case of a morphism of schemes and generic points of irreducible components. Here the result follows as generalizations lift along flat morphisms of schemes, see Morphisms, Lemma 29.25.9. $\square$


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